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The Alexander Wallace Spanier cohomology theory associates with an arbitrary topological space an abelian group. In this paper, an arbitrary topological space is associated with an R-module. The construction of the R-module is similar to the Alexander Wallace Spanier construction of the abelian group.

If G is a nite group and if X is a G-space, then a Bredon RO(G)-graded equivariantcohomology theory is dened on X. Furthermore, if X is a G-manifold, thereexists a natural Čech hypercohomology theory on X. While Bredon RO(G)-gradedcohomology is important in the theoretical aspects, the Čech cohomology is indispensablewhen computing the cohomology groups. The purpose of this dissertation is toconstruct an isomorphism between these two types of cohomology theories so that theinterplay becomes deeper between the theory and concretely computing cohomologygroups of classical objects. Also, with the aid of Čech cohomology, we can naturallyextend the Bredon cohomology to the more generalized Deligne cohomology.In order to construct such isomorphism, on one hand, we give a new constructionof Bredon RO(G)-graded equivariant cohomology theory from the sheaf-theoreticviewpoint. On the other hand, with Illman's theorem of smooth G-triangulation ofa G-manifold, we extend the existence of good covers from the nonequivariant tothe equivariant case. It follows that, associated to an equivariant good cover of aG-manifold X, there is a bounded spectral sequence converging to Čech hypercohomologywhose E1 page is isomorphic to the E1 page of a Segal spectral sequence whichconverges to the Bredon RO(G)-graded equivariant cohomology. Furthermore, Thisisomorphism is compatible with the structure maps in the two spectral sequences. So there is an induced isomorphism between two limiting objects, which are exactly theČech hypercohomology and the Bredon RO(G)-graded equivariant cohomology.We also apply the above results to real varieties and obtain a quasi-isomorphismbetween two commonly used complexes of presheaves.

If G is a nite group and if X is a G-space, then a Bredon RO(G)-graded equivariantcohomology theory is dened on X. Furthermore, if X is a G-manifold, thereexists a natural Čech hypercohomology theory on X. While Bredon RO(G)-gradedcohomology is important in the theoretical aspects, the Čech cohomology is indispensablewhen computing the cohomology groups. The purpose of this dissertation is toconstruct an isomorphism between these two types of cohomology theories so that theinterplay becomes deeper between the theory and concretely computing cohomologygroups of classical objects. Also, with the aid of Čech cohomology, we can naturallyextend the Bredon cohomology to the more generalized Deligne cohomology.In order to construct such isomorphism, on one hand, we give a new constructionof Bredon RO(G)-graded equivariant cohomology theory from the sheaf-theoreticviewpoint. On the other hand, with Illman's theorem of smooth G-triangulation ofa G-manifold, we extend the existence of good covers from the nonequivariant tothe equivariant case. It follows that, associated to an equivariant good cover of aG-manifold X, there is a bounded spectral sequence converging to Čech hypercohomologywhose E1 page is isomorphic to the E1 page of a Segal spectral sequence whichconverges to the Bredon RO(G)-graded equivariant cohomology. Furthermore, Thisisomorphism is compatible with the structure maps in the two spectral sequences. So there is an induced isomorphism between two limiting objects, which are exactly theČech hypercohomology and the Bredon RO(G)-graded equivariant cohomology.We also apply the above results to real varieties and obtain a quasi-isomorphismbetween two commonly used complexes of presheaves.

No description available.

nvestigando a estrutura dos sistemas elementares com simetria de Poincaré em 1 + 1 dimensões, devemos considerar o problema da eliminação das anomalias clássicas, que têm origem no segundo grupo de cohomologia não-trivial deste grupo dinâmico, gerando um termo de Wess-Zumino na ação da partícula relativística. Efetuamos a classificação geral de todos os sistemas elementares em 1 + 1 dimensões, em termos de co-órbitas, mostrando que existe um simplectomorfismo entre o espaço de fase reduzido da partícula e uma determinada co-órbita na álgebra de Lie dual à de Poincaré estendida. / While researching the structure of elementar systems with Poincaré symmetry in 1+1 dimensions, we must be concerned about the problem of elimination of the classical anomalies, which arise from the non-trivial second cohomology group of this dynamical group, generating a Wess-Zumino term in the relativistic particle action. We classify all elementary systems in 1+1 dimensions in terms of co-orbits, showing that there is a symplectomorphism between the reduced phase space of the particle and a certain co-orbit in the Lie algebra dual to the extended Poincaré one.

nvestigando a estrutura dos sistemas elementares com simetria de Poincaré em 1 + 1 dimensões, devemos considerar o problema da eliminação das anomalias clássicas, que têm origem no segundo grupo de cohomologia não-trivial deste grupo dinâmico, gerando um termo de Wess-Zumino na ação da partícula relativística. Efetuamos a classificação geral de todos os sistemas elementares em 1 + 1 dimensões, em termos de co-órbitas, mostrando que existe um simplectomorfismo entre o espaço de fase reduzido da partícula e uma determinada co-órbita na álgebra de Lie dual à de Poincaré estendida. / While researching the structure of elementar systems with Poincaré symmetry in 1+1 dimensions, we must be concerned about the problem of elimination of the classical anomalies, which arise from the non-trivial second cohomology group of this dynamical group, generating a Wess-Zumino term in the relativistic particle action. We classify all elementary systems in 1+1 dimensions in terms of co-orbits, showing that there is a symplectomorphism between the reduced phase space of the particle and a certain co-orbit in the Lie algebra dual to the extended Poincaré one.